new eng1060

prompt user for a number and checks whether it is -1, 0 or 1

%Prompting the user number = input('input a value'); %Applying a switch case on the user's input switch number case {-1} fprintf('number is negative one\n') case {0} fprintf('number is zero\n') case {1} fprintf('number is one\n') otherwise fprintf('number is none of the numbers\n') end

What is the pseudo code for bisection method?

1. Calculate the midpoint of [Xl, Xu] 2. Calculate f(Xr) and f(Xl) 3. Check subinterval for root, f(Xr)*f(Xl). Reset interval bounds 4. Repeat step 1 to 3, if f(Xr) is not close enough to zero

How many iterations are there in the following loop? k = linspace(4,3,20); for h = 1:length(k)-1 k(h) = k(h)^2 + k(h+1); clc end

19

What is the value of count after the following MATLAB commands? c = [6 1 4; 2 1 5; 3 7 1]; n = length(c); count = 0; for i = 1:n for j = 1:n if ( i==j & c(i,j)==1 ) count = count + 1; end end end

2

What is the value of a after running the following commands? sqr = @(x) x^2 a = sqr(5)

25

How many times is q being calculated in the inner most loop (x counter) when counter i=30 and counter k=-3? for i = 1:40 j = 4; for k = 5:-1:-5 s = 5 + j + k; for x = 5: -3: -9 q = -x*s; end end end

5

Benefits and limitations of switch

A "case" cannot use relational operators such as < or > Therefore to test for inequality use if statements, because cases can only contain discrete values The break command does not work in switch statements

A bracket root-finding method requires:

A lower bound and upper bound which brackets a root

What are breakpoints? What happens when your code hits break point?

Appear as red circles grey circlesindicate syntax error or unsaved m files conditional breakpoints appear as yellow circles CONTINUE: Continues running the code until next breakpoint STEP: Steps through single line of code STEP IN/OUT: Steps in and out of function files and loops RUN TO CURSOR: run the code to the current cursor position QUIT DEBUGGING: Exits debugging code

estimate total error

Calculatye sum of squared errors between fitting line and each data point.

What does the continue command do?

Commands after the continue command will not run

What is a characteristic of the secant method

Derivative is approximated using chord

What are the inputs and outputs of the newton raphson method?

INPUTS: A function to find the root A function that specifies the derivative A starting guess A precision we are prepared to accept OUTPUTS: The root of f(x)

When to use open methods

If we have no idea where the root might be Open methods usually converge faster, but can also diverge open methods: NEWTON RAPHSON SECANT MODIFIED SECANT

All open methods for root finding:

May diverge when approximating the next guess of the root Use of gradient to approx the next guess of the root Is sensitive to the shape of the function

difference between secant and modified secant method

Modified secant method you only need one point to start. Need to make 2 function evaluations each step.

Method of False Position

Need 2 values of x with opposite signs Draw a chord between the 2 points Where the chord crosses the axis is root estimate SOME IMPORTANT POINTS: You must bracket the root otherwise false position cannot choose subinterval to use next. If there is more than one root which you find is down to luck

difference between newton raphson and secant method

Need to start with 2 guesses they do not have to be bracketed best to make them fairly close to each other

Which method results in divergence?

Newton Raphson: And the exact derivative of the function is required

Whatis the characteristic of the modified secant method

One initial guess and a perturbation is required to begin the algorithm

How to validate your code with pseudo code

Run your code to ensure that there are no syntax errors. Address any errors that are encountered Justify your answers. Does it look correct? Use different toold (plotting) to validate your answers if the numbers look incorrect bgein debugging

Standard deviation and Variance

SD = residuals variance is SD^2

What is a switch statement?

Similiar to if statements Checks one variable against cases (can contain multiple values)

function handles

So you dont need to create and manage too many funtion files. Can have many input arguments but can only return one output. syntax: function_handle = @(inputs) <commands>

Is it possible to terminate a loop or skip an iteration?

The break command terminates both for and while loops commands after the break command will not run In a nested loop, a break command will only break out of a single loop A break command should only be used to handle unexpected situations A well coded program will not require break commands

Airthmetic mean of a set of n data values:

The sum of individual data points (yi) divided by the number of points.

What is debugging? What are good programming practices that will reduce bugs?

The task of identifying and correcting errors (bugs) in code. Functions comments indenting pseudocode descriptive variable names

What is the errror at a point "i"

The vertical distance of the point i from the line of best fit.

difference between underflow and overflow

Underflow for double precision is smaller absolute value than 2.22507e-308, overflow is larger absolute value than 1.79769e+308. ... Overflow will create an infinity.

How to find where a function crosses the x axis.

Use matlabs fzero() fn = @(x) -5*x.^5 + 400*x.^4 x0 = zero(fn, 75)

Variables are automatically stored in which 2 workspaces

Variables created using the command window or m files are stored inside the base workspace Variables created inside a function are stored inside the function workspace

What will cause a while loop to end?

When the while condition becomes false

When is a while loop used?

When you dont know how many iterations you require. Instead you have a condition you want to meet. while <condition> <commands> end

What is the value of A after the following MATLAB commands? A = eye(3, 3); for y = 1:2:4 A(y, y) = 0; end

[ 0 0 0 ; 0 1 0 ; 0 0 0 ]

x = 1; y = 1; while y < 4; z(x) = 2 .* y; x = x + 1; y = y + 2; end

[2 6]

Finding the row and the column of the maximum number in the matrix: A = [1 2 3 4 ; 5 6 7 6; 5 4 3 2]

[r,c] = find(A ==max(max(A))) r = 2 c = 3

Which of the following is not a variable class (type)?

anonymous function

Binary numbers

built in function Converts between binary and decimal (base - 10) numbers syntax: number = bin2dec('string') dec_num = bin2dec('111') % 2^2 + 2^1 + 2^0 = 7 string = dec2bin(number) bin_num = dec2bin(7)

What is the value of a after the running the following commands? sqr = @(x) x^2 a = sqr([1:5])

error

What condition needs to be satisified to intiate a CLOSED ROOT FINDING METHOD

f(Xl) * f(Xr) > 0

Print the numbers 1, 3, 5... 11

for counter = 1:2:11 fprintf('counter is %d\n, counter'); end

difference bteween for loop and while loop

for loop: when you know how many times to repeat a code while loop: when you know what condition to meet.

print numbers 1^2, 2^2....10^2

for r = 1:10 fprintf('%d squared is%d\n', r, r^2) end

print 5 random numbers

for u = 1:5 fprintf('random number: %f\n', rand) end

Print 10 by 10 grid that count from 1 to 10, 2 to 20, 3 to 30 etc. numbers in the first row seperated by 1. numbers in the second row seperated by 2. numbers in the third row seperated by 3.

for x = 1:10 for y = 1:10 fprintf('%3d ', y*x); end fprintf('\n') end

Store values in a matrix instead of printing

for x = 1:10 y = 1:10 M(x,y) = x*y; end end % x is for rows % y is for columns

newton raphson method code

function [root, iter] = newraph(f, df, xi, precision) % [root, iter] = newraph(f, df, xi, precision) % General purpose algorithm for Newton-Raphson Method % % INPUTS: % - f: function handle of the equation to be solved % - df: function handle of the derivative of the equation to be solved % - xi: the initial guess / the next guess x_i_+_1 % - precision: stopping criteria determined by the user % OUTPUT: % - root: the root of the equation % - iter: total iteration taken % Estimate 1st iteration of xi1 and initialise iteration count iter = 1; %estimate the root for the equation xi1 = xi - (f(xi)/df(xi)); % Iteration for Newton-Raphson method starts %Does the function value satisfy precision? while abs(f(xi1)) > precision % Increment the iteration count by 1 iter = iter + 1; % updating variables %the old root becomes the new initial guess xi = xi1; % recalculating xi1 xi1 = xi - (f(xi)/df(xi)); end % The final xi1 value is the root root = xi1;

code for bisection method

function [root,iter] = bisection(f, xl, xu, precision) % [root,iter] = bisection(f, xl, xu, precision) % INPUTS: % - f: function handle of the equation to be solved % - xl: lower limit of the initial guess/bracket % - xu: upper limit of the initial guess/bracket % - precision: stopping criteria determined by the user % OUTPUT: % - root: the root of the equation % checking if bounds are appropriate if f(xu) * f(xl) > 0 error('bounds are innappropriate'); end % Estimate 1st iteration of xr and initialise iteration count iter = 1; xr = (xl+xu)/2; % Check if f(xr) is close enough to zero while abs (f(xr)) > precision %what do we want to satisfy? % checking subinterval for root if f(xl) * f(xr) < 0 xu = xr; else xl = xr; end % Recalculate xr and update iteration count iter = iter + 1; xr = (xl+xu)/2; end % The final xr value is the root root = xr;

Method of False Position code

function [root,iter] = falseposition(f, xl, xu, precision) % [root,iter] = falseposition(f, xl, xu, precision) % INPUTS: % - f: function handle of the equation to be solved % - xl: lower limit of the initial guess/bracket % - xu: upper limit of the initial guess/bracket % - precision: stopping criteria determined by the user % OUTPUT: % - root: the root of the equation % checking if bounds are appropriate if f(xu) * f(xl) > 0 error('bounds are innappropriate'); end % Estimate 1st iteration of xr and initialise iteration count iter = 1; xr = (f(xu)*xl - f(xl)*xu)/(f(xu)-f(xl)) ; % Check if f(xr) is close enough to zero while abs (f(xr)) > precision %what do we want to satisfy? % checking subinterval for root if f(xl) * f(xr) < 0 xu = xr; else xl = xr; end % Recalculate xr and update iteration count iter = iter + 1; xr = (f(xu)*xl - f(xl)*xu)/(f(xu)-f(xl)); end % The final xr value is the root root = xr;

function header declaration

function outputs = function_name(inputs)

Bracketed methods with bounds [a, b] can only begin if:

function values at a and b are of opposite signs

What is matlabs inbuilt function to finding the root

fzero(<function_handle>, <initial_guess>)

Which variables can exist and be shared in both workspaces?

global variables syntax: global <variable name> <variable name> = <value>

decrease n by 1 until <=1

n = 5 while n>1 dis(n); n = n-1; end

Find where the maximum number is in the vector A = [1 3 5 5 3 2 ]

new = find(A ==max(A)) new = 3 4 %The 3rd and 4th number are the highest

pre allocate matrices with the zeros function

time = [1:10; 11:20; 21:30; 31:40]; [r,c] = size(time); speed = zeros(r,c);

The command -999^-999 results in:

underflow

The command 999^-999 results in:

underflow

Major pseudo components

variables assignment input/ouput condition repetition

continue to multiply x by 8 until x^2 is larger than 123456789

x = 1; iter = 0 while x^2<123456789 iter = iter+1 x = 8*x fprintf('iteration=%i, x=%i, x^2=%i\n', iter,x,x^2)